Short-circuit testing model for stationary induction apparatuses

ABSTRACT

A stationary induction apparatus in which the number of natural vibrations of a winding formed by winding an insulated conductor by a plurality of turns about a leg portion of an iron core is shifted from the power source frequency and the frequency which is double the power source frequency so as to prevent the winding from resonating with the latter freqeuncies and to reduce the electromagnetic force generated in the axial direction of the winding due to a short-circuit current which may flow into the apparatus winding and due to the leakage flux occurring in the apparatus, thereby to obtain a mechanically strong winding structure.

United States Patent [191 Hori et al.

[ Jan. 15, 1974 1 SHORT-CIRCUIT TESTING MODEL FOR STATIONARY INDUCTIONAPPARATUSES [75] Inventors: Yasuro Hori; Kiyoto Hiraishi;

Tadashi Kiuchi; Yoshitake Kashima, all of Hitachi, Japan [73] Assignee:Hitachi, Ltd., Tokyo, Japan [22] Filed: July 14, 1971 [21] Appl. No.:162,601

Related U.S. Application Data [62] Division of Ser. No. 794,862, Jan.29, 1969,

abandoned.

[30] Foreign Application Priority Data .Ian. 31, 1968 Japan 43/5419 [52]US. Cl 336/181, 336/60, 336/197, 336/212 [51] Int. Cl. H011 27/30 [58]Field of Search 336/170, 171, 183, 336/212, 214, 215, 197, 180, 181,182, 60

[56] References Cited UNITED STATES PATENTS 1,641,659 9/1927 Brand336/183 X 0 SECT/O/V) Primary Examiner-Thomas J. Kozma Attorney-Craig &Antonelli [57] ABSTRACT A stationary induction apparatus in which thenumber of natural vibrations of a winding formed by winding an insulatedconductor by a plurality of turns about a leg portion of an iron core isshifted from the power source frequency and the frequency which isdouble the power source frequency so as to prevent the winding fromresonating with the latter freqeuncies and to reduce the electromagneticforce generated in the axial direction of the winding due to ashort-circuit current which may flow into the apparatus winding and dueto the leakage flux occurring in the apparatus, thereby to obtain amechanically strong winding structure.

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SHEET 7 BF 9 F/G /70 F/G /7b INVENTOR5 ATTORNEYS SHORT-CIRCUIT TESTINGMODEL FOR STATIONARY INDUCTION APPARATUSES This is a divisional ofApplication Ser. No. 794,832, filed Jan. 29, 1969, now abandoned.

BACKGROUND OF THE INVENTION 1. Field OfThe Invention This inventionrelates to stationary induction apparatus.

2. Description Of The Prior Art Stationary induction apparatus such astransformers and reactors of large capacity employed in power circuitscomprise essentially an iron core, windings disposed to surround theiron core, and electrical insulators for electrically insulating thewindings from the iron core as well as one winding from the other.

Commonly, the electrical insulator described above is formed from papercomprised essentially of fibrous materials. For example, Kraft paper,Manila paper or pressboard is generally used to form such members asinsulating coverings for conductors of individual coils constituting thewiring, insulating cylinders disposed between the windings, insulatingrings disposed above and beneath the windings and interposed between thecore fastening members and the yokes of the iron core, and inter-coilduct pieces disposed radially between the coils in suitably spacedrelation from each other.

When short-circuit trouble occurs in the power transmission system forthe stationary induction apparatus formed from materials as describedabove, an excessively large short-circuit current flows through therelated winding of the apparatus, and this short-circuit currentcooperates with the radial component of leakage flux in the apparatus toimpart an excessively large electromagnetic force to the winding in itsaxial direction. the axial electromagnetic force acts to createalternate compression and vibration between the coils in the winding andthe inter-coil duct pieces as well as between the winding and theinsulating rings. Due to the vibration created in this manner, gaps areproduced between the coils and the inter-coil duct pieces as well asbetween the upper and lower ends of the winding and the insulatingrings, and because of the presence of the gaps, an excessively largeimpact is imparted between the coils and the inter-coil duct pieces aswell as between the upper and lower ends of the winding and theinsulating rings in the succeeding period of vibration.

It is known that the winding in the electrical apparatus of this kindhas the natural vibrations or the natural frequencies of the firstorder, second order and third order which lie in the vicinity of 30cycles per second, 70 cycles per second and 120 cycles per second,respectively. Thus, these natural frequencies of the winding are veryclose to the power source frequency, 50 hertz or 60 hertz, and itsdouble frequency, 100 hertz or 120 hertz. Since the natural frequenciesof the winding are thus very close to the power source frequencies whichimpart vibration to the winding, the displacement due to the vibrationis quite large, and in some cases, resonance takes place to furtherenlarge the above displacement. As a result, the winding is subject topermanent plastic deformation due to the abovedescribed impact ordisplacement until finally it is broken.

The inter-coil duct pieces and the insulating rings described above musthave the function of insulating the individual coils from each other andinsulating the windings from the earth, and at the same time, thefunction of mechanically holding the individual coils and the entirewindings. However, these inter-coil duct pieces and insulating ringshave a certain limit in their mechanical strength because of the factthat they consist essentially of paper material. As a result, theseinter-coil duct pieces and insulating rings are quite weak against theelectromagnetic force and impact described above and are liable to moveout of their predetermined position to be easily broken down. Thecollapse of the duct pieces and insulating rings would further promotethe deformation of the winding.

SUMMARY OF THE INVENTION It is therefore an object of the presentinvention to provide a stationary induction apparatus having a windingstructure which does not easily deform even if a short-circuit currentoccurring in an associated system may flow into the winding.

Another object of the present invention is to provide a model for astationary induction apparatus which verifies economically the fact thatthe windings thereof have a sufficient strength against anelectromagnetic force applied thereto.

The present invention contemplates the provision of a stationaryinduction apparatus having a mechanically stable and strong winding inwhich, on the basis of the result of analysis of the electromagneticforce applied to the winding in the case of short-circuiting, the numberof natural vibrations of the winding is so set that it is higher thanthe frequency of a power source and is suitably shifted from the valuewhich is double the power source frequency in order to thereby suppressthe axial vibrational compression and displacement of the winding due toimpartation thereto of the electromagnetic force.

In accordance with the present invention, the winding itself has arigidity greater than that of the insulating ring so that the winding issufficiently mechanically stable and strong. The total pressurereceiving area of the inter-coil duct pieces disposed between theadjacent coils is made larger than the pressure receiving area of theinsulating ring in the axial direction of the winding so that thewinding can be made quite strong mechanically. Further, the ampereturndistribution in the axial direction of the winding may be varied so thatthe electromagnetic force developed in the axial direction of thewinding has a distribution with a mode of the fourth or higher order inorder thereby to attain the effect similar to that which is attained byincreasing the number of natural vibrations of the winding itself.

Moreover, in accordance with the present invention, it is possible toobtain a test model by which the practical investigation of the mannerof occurrence of the above-described electromagnetic force can easily becarried out. By virtue of the fact that the test model may be soconstructed as to eliminate a part of the iron core and windings withoutin any way losing the equivalency to a proper stationary inductionapparatus, the test of this kind which has heretofore been considereddifficult to execute from an economical point of view can inexpensivelybe carried out.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. I is a vertical sectional sideelevational view showing part of the winding structure wound around aniron core in the stationary induction apparatus of the presentinvention.

FIG. 2 is a sectional view taken on the line II-II in FIG. 1.

FIG. 3 is a diagrammatic illustration of leakage flux distribution in atwo-winding transformer.

FIGS. 4a, 4b and 4c are diagrammatic illustrations of the distributionof electromagnetic forces developed in the axial direction of thewinding when short-circuit takes place in an associated system.

FIG. 5 is a diagrammatic illustration of an equivalent vibration circuitin which the apparatus winding is represented by a concentrated constantsystem.

FIG. 6 is a diagrammatic illustration of an equivalent vibration circuitin which the apparatus winding is represented by a distributed constantsystem.

FIG. 7 is a diagrammatic illustration of the coordinates of vibration inthe distributed constant systern.

FIGS. 8 through 12 are vertical sectional views showing various forms ofthe winding structure according to the present invention.

FIGS. 13a and 1312 are diagrammatic illustrations of the mode ofelectromagnetic force distribution and the form of normal vibrations ofvarious orders, respectively.

FIGS. 14a 14b are diagrammatic illustrations of the distribution of theparticipation factor Ki corresponding to the mode of various orders.

FIGS. 15a, 15b through 17a, 17b are diagrammatic illustrations ofvarious arrangements of the low-voltage winding according to the presentinvention and corresponding distributions of the electromagnetic forcedeveloped in the axial direction of the winding, respectively.

FIG. 18a is a diagrammatic view showing the structure of a transformermodel according to the present invention for testing for theelectromagnetic force developed in the axial direction of the winding.FIGS. 18b and 180 are sections taken on the lines b-o and c.-o in FIG.18a, respectively.

FIGS. 19 is a diagrammatic illustration of one example of thedistribution of the electromagnetic force in the axial direction of thewinding in the transformer model.

FIGS. 20a and 2012 are diagrammatic illustrations of another form of thetransformer model and the distribution of the electromagnetic force inthe axial direction of the winding, respectively.

FIG. 21 is a diagrammatic illustration of a further form of thetransformer model for verifying the distribution of the electromagneticforce in the axial direction of the winding.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring to FIGS. 1 and 2showing the internal structure of a two-winding transformer, a pair ofconcentric windings 4 and 7 are disposed to surround an iron core I. Asis commonly known, the transformer iron core I comprises a leg portionla which is assembled by laminating a multiplicity of steel sheets andyoke portions lb which are assembled by laminating a multiplicity ofconcentric iron sheets and are magnetically coupled to the top andbottom of the leg portion la. The leg portion la and yoke portions 1bare fastened together by suitable means. An insulating cylinder 2 ofpressboard tits on the leg portion la of the iron core 1. A plurality ofrod-like insulating spacers 3, commonly made of pressboard, are bondedto the outer peripheral surface of the insulating cylinder 2 in suitablyparallelly spaced relation from each other.

The low-voltage winding 4 of the transformer is disposed to surround theleg portion 1a of the iron core l with the spacers 3 interposedtherebetween and comprises a multiplicity of coils 4 4 4 4,, 4 t -2, 4,,and 4,, wound in its axial direction. These coils are successivelyelectrically connected with each other to form a discal winding, orinsulated conductors 4a constituting the individual coils are bundledand continuously wound to form a helical winding. A plurality ofinter-coil duct pieces 5,, 5 5 5,, 5 5 and 5 are radially disposedbetween the adjacent coils in suitably spaced relation from each other.The inner end of each duct piece situated on the side of the innerperiphery of the coil makes a dovetail joint with the correspondinginsulating spacer 3. Insulating layers 6 comprising a plurality ofconcentric insulating cylinders are disposed to surround the outerperipheral surface of the low-voltage winding 4 so that oil gaps formedbetween the adjacent cylinders and the insulating cylinders constitutethe main insulation.

The high-voltage winding 7 is disposed to surround the main insulationand is commonly in the form of a discal winding or a cylindricalwinding. In the illustrated embodiment, the high-voltage winding 7 is inthe form of a discal winding. The high-voltage winding 7 comprises analternate arrangement of coils 7 7 7,, and 7,, formed by windinginsulated conductors 7a from the outer periphery toward the innerperiphery of the winding and coils 7 7 7 and 7 formed by winding theinsulated conductors 7a from the inner periphery toward the outerperiphery of the winding. A plurality of inter-coil duct pieces 8 8 88,, 8, and 8 are radially disposed between the adjacent coils insuitably spaced relation from each other. The inner end of each ductpiece situated on the side of the inner periphery of the coil makes adovetail joint with a corresponding one of insulating spacers 9parallelly disposed on the outermost layer of the main insulation.

Shielding rings 10 are disposed on the upper and lower ends of thelow-voltage winding 4. Shielding rings 11 are disposed on the upper andlower ends of the high-voltage winding 7. These shielding rings 10 and11 are made by covering a conductor with an insulating tape. Insulatingrings 12 are interposed between the yoke portions lb of the iron core 1and the shielding rings 10. Insulating rings 13 are interposed betweenthe yoke portions lb of the iron core 1 and the shielding rings 11.These insulating rings 12 and 13 support insulatingly the low-voltagewinding 4 and the high-voltage winding 7 in their vertical direction.

FIG. 3 shows the manner of distribution of leakage flux in a two-windingtransformer as described above. With the leakage flux distribution assuch, when a short-circuit current flows into one of the windings, alarge electromagnetic force is developed in the internally disposedlow-voltage winding 4 and the externally disposed high-voltage winding7. Such an electromagnetic force is commonly of the order of severalhundred tons.

FIG. 4 shows the distribution of the electromagnetic force developed inthe axial direction of the lowvoltage winding 4. In FIG. 4a, it will beseen that electromagnetic forces f ,f ,f ,f,, .f fl f .f,, and f aredeveloped in the respective coils 4,, 4 4 4 4,, 4 4,], 4,, and 4,, whenthe shortcircuit current flows into the winding. The electromagneticforces developed in the coils have a distribution as shown in FIG. 4bfrom which it will be seen that the electromagnetic forces aresubstantially opposite to each other on opposite sides of the centralportion of the winding 4 and are maximum at the upper and lower ends ofthe winding 4 where the radial component of the leakage flux is large.However, the electromagnetic force developed in the individual coil istransmitted intact to the adjacent coils since the inter-coil ductpieces 5,, 5,, are generally merely interposed between the adjacentcoils. As a result, the electromagentic forces described above arecombined together and the resultant force produces a maximum compressionat the central portion of the winding 4. the resultant force isdistributed in the form of if as shown by the solid line in FIG. 4c inwhich the vertical axis represents the axial length L of the winding 4and the horizontal axis represents the developed mechanical force F.

In an attempt to analyze the electromagnetic force developed in theaxial direction of the winding, the inventors investigated the vibrationsystem of the winding. In the analysis, a concentrated constant systemas shown in FIG. 5 is replaced. by a distributed constant system asshown in FIG. 6.

Suppose now that each coil in the winding has weight a, each inter-coilduct piece has a spring constant k, and each insulating ring has aspring constant K. Then, an equivalent vibration circuit according tothe concentrated constant system will be as shown in FIG. 5. However,the winding of a stationary induction apparatus such as a transformer orreactor includes many particles due to the fact that the number of coilsranges from several tens to several hundreds. Thus, it is verytroublesome to analyze the concentrated constant system in such acomplicated form. Therefore, the inventors replaced the vibration systemof the winding by a distributed constant system by noting the fact thatthere are very many coils. The error due to regarding a concentratedconstant system as a distributed constant system is substantiallynegligible as a matter of practice. This is because the naturalvibrations of the first order in the winding of the electrical apparatusof this kind lie generally in the vicinity of 20 to 50 cycles per secondand the natural vibrations of the hundredth order are the windingconsisting of, for example, one hundred coils is far higher than thepower source frequency of 50 Hz or 60 Hz, and thus the errorattributable to the distributed constant system is negligible even ifthe number of natural vibrations of the hundredth or higher order may beintroduced in the systern.

In FIG. 6 there is shown an equivalent vibration circuit of thedistributed constant system based on the above way of thinking. For theconversion of the concentrated constant system into the distributedconstant system, the following equations may only be satisfied;

a a' l/ d L1 d'g' where E,,: modulus of longitudinal elasticity of thedistributed constant system in kg/cm k,,: spring constant of theinter-coil duct piece in kg/cm X mean pitch of the coil in cm 8,pressure receiving area of the coil in cm Q density of the distributedconstant system in w mean weight of the coil in kg g acceleration ofgravity in cm/S FIG. 7 shows the co-ordinates of the distributedconstant system under consideration, in which the X-axis lies in thedirection of height of the winding 4. Now, the response of the vibrationsystem in FIG. 7 when an external force F is imparted to an arbitrarilyselected point x al will be obtained. The solution in the steady statewill be obtained supposing that the external force F F e Since the waveequation holds in the winding shown in FIG. 7, the following equation isobtained when the displacement in the axial direction is y(x, t):

where C is the velocity of sound V E /p in cm/sec. As is commonly known,the solution in the steady state when 0 x al is given by and thesolution in the steady state when al 5 x g 1 is giveny by )2 (Ce D -J wlcr d d 1 (x, l yl (L By inserting y (x,t) in the equation (4) into theequation (6), we obtain the following equation:

jm/c EdSd (A B) K (A B) At x al, the displacement is continuous and weobtain the following equation:

yl (M) )2 (M) Similarly, at x =al, the pressure is continuous and weobtain the following equation:

By inserting y (x,t) in the equation (4) and y (x,t) in the equation (5)into the respective equations (8) and (9), the following equations canbe obtained:

5,5,, (Ae We Be'j /c al) F,

...j E s 01c al D j rule 111) Atx=l,

By inserting y (oc,t) in the equation (5) into the equation (12), weobtain the following equation:

The factors A, B, C and D can be sought from the equations (7), (10),(ll) and (13). Suppose that (wE S /cK) 'y' and by inserting 'y' and 8'into the equations (7), (10), (11) and (13), we obtain the followingequations, respectively:

considered to be equivalent to each other when 7' and 6' are the samefor both.

Suppose that y is the quotient obtained by dividing the equation (l4) bythe equation (15) and 8 is the product obtained by multiplying theequation (14) by the equation (15), then we obtain the followingequations:

Y d d/ 5 (w IE S /C K) (w MlKg) where M is the total weight of thewinding in kilograms. It will be seen that the quantity 7 is independentof the frequency.

Consider the physical meaning of the equation (20). It is apparent thaty is the ratio of the spring constant E S /l when the entire winding isregarded as a spring to the spring constant K of the insulating ring. Inother words, 7 may be considered as representing the ratio of the totalheight of the insulators disposed between the coils to the height of theinsulators including the insulating rings which are disposed outside ofthe coils.

Next, consider the physical meaning of the equation (21). By multiplyingthe denominator and numerator of the equation (21) by a certaindisplacement u, we obtain the following equation:

(2 The numerator of the equation (22) represents the inertia force whenit is supposed that the winding moves as an integral body, while thedenominator of the equation (22) represents the spring force of theinsulating ring. In other words, 5 in the equation (22) may beconsidered as representing the ratio of the weight of the winding to thespring constant of the insulating ring.

Although the exciting force is distributed in the axial direction of thepractical winding, the equivalency can be maintained in the presentanalysis since the relation obtained in the above discussion holds atany axial position and the value may be integrated over the entireforce.

Finally, we will investigate as to how the parameters 7 and 8 should bedesigned in order to reduce the vibration of the actual transformerwinding and to reduce the force imparted to the coils and the upper andlower supports for the coils.

It will be known from the equations (4) and 5) that the displacement ofthe coils is related with the factors A, B, C and D. The equation 18)among the equations (16) through (19) determines the factors A, B, C andD, and it is known that the displacement of the coils is proportional toF /yK. In other words, the relation can be expressed as by the springconstant K of the insulating ring as follows:

upper and lower insulating rings is a matter of special consideration.Now, we will investigate the equation (24). From the equations (20) and(21 the following equation is obtained:

W V tEdsmm d d/ H wee/ K (2 Therefore, the equation (24) can beexpressed as 5 CKFO fll (26) wE S 1 (E (1S1! (0S4 [IE 1 On the basis ofthe above analysis, it was found that the desired reduction in thedisplacement of the winding when a short-circuit current flowstherethrough and .the desired reduction in the force imparted to the WtwEdsm m 8. eal/K) Therefore, an effective practical arrangement for thewinding so that it is sufficiently resistive to the electromagneticforce is such that the winding and the upper and lower supports thereforhave a unitary structure in which the upper and lower supports are softand the winding is rigid. The number of natural vibrations of thewinding can be increased by making the winding rigid in this manner. Theincrease in the number of natural vibrations of the winding is quitepreferable in that there is utterly no chance for the winding toresonate with the power source frequency or with the frequency which isdouble the power source frequency. The resonance is objectionable sinceit results in an increase in the amplitude of vibration, hence in anincrease in the displacement of the winding.

The present invention will more practically be described on the basis ofthe above elucidation.

The spring constant K of the insulating rings disposed above and beneaththe winding is given by K I/ l) 1 where S, is the pressure receivingarea of the insulating ring, I, is the height of the insulating ring andE is the modulus of longitudinal elasticity of the insulating ring.

The modulus of longitudinal elasticity E, of the winding and thepressure receiving area S of the winding can be calculated from theshape, number, modulus of longitudinal elasticity and other factors ofthe inter-coil duct pieces included within the winding. The modulus oflongitudinal elasticity E of the winding is given by the followingequation:

E =1 ,nli E where l height of the winding including the inter-coil ductpieces h height (thickness) of each inter-coil duct piece n number ofstages of the inter-coil duct pieces in the axial direction of thewinding E modulus of longitudinal elasticity of the inter-coil ductpiece On the other hand, the pressure receiving area 8,, of the coil isgiven by the following equation:

S1] mS where m; number of the inter-coil duct pieces interposed betweena pair of coils S; pressure receiving area of each inter-coil duct pieceIn order to increase the values of'y and 6 described pre viously, thevalue of K in the equation (27) may be decreased or the value of E inthe equation (28) may be increased.

The above relations may be summarized as follows:

i. The modulus of longitudinal elasticity E of the inter-coil duct pieceshould be increased.

2. The modulus of longitudinal elasticity E of the insulating ringshould be decreased.

3. The number m of inter-coil duct pieces interposed between a pair ofcoils should be increased.

4. The pressure receiving area S of each inter-coil duct piece should beincreased.

5. The pressure receiving area S, of the insulating ring should bedecreased.

6. The height l of the insulating ring should be increased.

7. The height 1 of the winding should be increased.

8. The height (thickness) h of each inter-coil duct piece should bedecreased.

The item (3) among the above-described items is undesirable because itis difficult to secure an enough space for the passage of the coolingmedium for the winding as the number of the inter-coil duct piecesincreases. The item (6) is also undesirable in that an increase in theheight of the insulating ring beyond the required insulation distance ofthe winding to the! ground results in a large size of the iron core,hence bulkiness of the apparatus as a whole. The item (7) is alsoundesirable in that the apparatus as a whole becomes bulky in size. Theitem (8) is subject to a limitation because there must be a sufficientinsulation distance between the adjacent coils.

It is therefore preferable to put the items l (2), (4)

cording to an experiment, good results were obtained when the modulus oflongitudinal elasticity of the intercoil duct pieces was substantiallyfive times or more than that of the insulating rings.

For example, the insulating ring may be made from conventionalpressboard which is manufactured according to Grade PB-1 or PB-2 ofJapanese Industrial Standards C2305, and the inter-coil duct piece maybe made from high-density highly-compressed pressboard prepared bystrongly heating pulp, forming the fibres into paper, applying apressure which is several times the prior pressure to the paper at ahigh temperature to compress the same and drying the compressed product.

The properties of the conventional pressboard according to Grade PB-2are compared with the properties of the high-density, highly-compressedpressboard and the results are given in the following table:

High PB-Z density (prior highlyproduct) compressed pressboard Density O.9-l.l l.l5l.3 Oil absorption rate 20% Deflection by compression (1.00

kg/cm*) 2.5% 0.2% Dry shrinking rate Below 8% Below 6% (thickness)Longitudinal Tensile direction Above 6.0 Above 10.0 strength (kg/mm)(with thickness of Lateral 3.2 mm) direction Above 2.5 Above .0

(kg/mm) Materials having a modulus of longitudinal elasticity largerthan that of the pressboard Grade PB-l or PB-2 include electricalinsulating sheets of phenol resin or epoxy resin and reinforced wood.Good results can also be obtained when these materials are employed toform the inter-coil duct piece. Since these materials are satisfactoryin their corona suppression property in the oil, they can very easily beincorporated in the transformer.

Short-circuit current was made to flow through the transformer windingwhose inter-coil duct pieces were made from the high-density,highly-compressed pressboard described above and whose insulating ringswere made from the conventional pressboard Grade PB-Z. Test resultsproved that the electromagnetically produced mechanical force 2] wasdistributed in a manner as shown by the two-dot chain line in FIG. 40and the resultant maximum compressive force imparted to the windingcould be made far smaller than the force appearing in the prior artsystem. Thus, the winding itself can be made sufficiently strong againstthe shortcircuit current by reducing the mechanical force developed inthe winding.

In the present invention, the items (4) and (5) described previously maybe adopted in order that the inter-coil duct pieces disposed between thecoils have a total pressure receiving area which is larger than that ofthe insulating ring.

FIGS. 8 through 12 illustrate several embodiments of the presentinvention in which the pressure receiving area of the inter-coil ductpiece is increased relative to that of the insulating ring so as toincrease the modulus of longitudinal elasticity of the winding withoutreducing the cooling area of the winding. Referring to FIG. 8,inter-coil duct pieces 17 are interposed between coils I6 16 16 16 I6forming a transformer winding 16. Each inter-coil duct piece 17 has aU-like sectional shape and comprises portions 17a and 17b disposedopposite to the upper and lower surfaces of the corresponding coil and aportion 17c connecting between the portions 17a and 17b at the inner orouter periphery of the coil. According to this embodiment, theinter-coil duct piece 17 has a quite large pressure receiving area whichincludes the areas of the portions 17a and 17b plus the cross-sectionalarea of the connecting portion 17c.

Embodiments shown in FIGS. 9 through 12 comprise an insulating cylinder18 disposed to surround a winding 16, and inter-coil duct pieces 17disposed between the insulating cylinders 2 and 18 in such a manner thateach inter-coil duct piece 17 covers the outer peripheral surface aswell as the inner peripheral surface of the corresponding coil. In thecase of the embodiment shown in FIG. 9, a cut 17d is provided in theouter peripheral portion of the duct piece 17 so that the duct piece 17is urged to open at this cut when it is fitted on the correspondingcoil. In the case of the embodiment shown in FIG. 10, oppositely alignedcuts 17d and l7e are provided in the inner and outer peripheral portionsof the duct piece 17 so that the duct piece 17 can be split into upperand lower halves. In the case of the embodiment shown in FIG. 11,diagonally opposite cuts 17d and 17e are provided in the inner and outerperipheral portions of the duct piece 17 so that the duct piece 17 canbe split into upper and lower L-shaped halves.

I In the case of the embodiment shown in FIG. 12, two

L-shaped duct pieces 17f similar to that shown in FIG. 11 are combinedwith a T-shaped duct piece 17g to surround a pair of coils.

The embodiments shown in FIGS. 9 through 12 are advantageous over theembodiment shown in FIG. 8 in that the duct piece has an increasedeffective pressure receiving area and thus the modulus of longitudinalelasticity of the winding can be further increased. It will be apparentfor those skilled in the art that the duct pieces 17 in any one of theembodiments shown in FIGS. 8 through 12 are radially disposed betweenthe coils in the radial direction of the latter in suitably spacedrelation from each other as in the case of the embodiment shown in FIG.2 so as to define a cooling passage.

According to the embodiments shown in FIGS. 8 through 12, part of themechanical forces F 1 through f which have heretofore been distributedto the winding as shown in FIG. 4a are carried by the inter-coil ductpiece 17, and as a result, the resultant force acting upon the entirewinding can be reduced in a manner as shown by the curve if in FIG. 40like the preceding embodiment.

In the above description, the number of natural vibrations of thewinding is increased as a result of the analysis of the vibration systemof the winding. In an alternative arrangement, suitable electomagneticmeans may be employed to vary the vibration mode of the winding for thesame purpose of increasing the number of natural vibrations of thewinding so as to similarly reduce the displacement of the winding and todecrease the force imparted to the insulating rings disposed above andbeneath the winding.

In the case of a transformer, for example, its electromagnetic forcedistribution takes generally the form of the second order correspondingto the distribution of the radial components of the leakage flux.Therefore, the vibration mode of the second order corresponding to thismanner of distribution is most liable to appear. Since the number ofnatural vibrations in the case of the vibration mode of the second orderlies in the vicinity of 70 cycles per second as described previously andis thus close to the power source frequency, the displacement due tovibration becomes correspondingly greater.

Now, a participation factor Ki is used to define the relation betweenthe electromagnetic force distribution and the mode of natural vibrationof respective orders. The participation factor Ki represents the rate atwhich the electromagnetic force distribution participates in thevibration mode of the ith order and can be given by the followingequation:

above relation. FIG. 13a shows the mode of the electro-- magnetic forcedistribution F(x) and FIG. 13b shows the normal mode of vibration of theith order, for example, those of the first order, second order andfourth order. In FIGS. 13a and 13b, x represents the axial length of thewinding. In FIG. 13a, the mode shown by the solid line represents theelectromagnetic force distribution of the second order in a conventionaltransformer, while the mode shown by the one-dot chain line representsthe electromagnetic force distribution of the fourth order which isimproved in accordance with the present invention.

Since the vibration mode tends to be induced in relation to theelectromagnetic force distribution, a vibration mode of higher order canbe developed when the electromagnetic force distribution is shifted toits higher order. On the basis of he above fact, the vibration mode canbe shifted to a number of natural vibrations of higher order by varyingthe electromagnetic force distribution while keeping the numbers ofnatural vibrations of various orders unchanged. In such an arrangement,the electromagnetic forces cancel each other at the numbers of naturalvibrations of lower orders, and the displacement of the winding can bemade corresponding smaller. This is equivalent to the effect as when thenumber of natural vibrations of the winding is increased.

In FIG. 14a there is shown the state in which the number of naturalvibrations of the winding itself is shifted to a higher order withoutvarying the electromagnetic force distribution as well as theparticipation factor Ki. The solid curve in FIG. 14a represents thedistribution of the participation factor Ki in a conventionaltransformer and it will be apparent that the participation factor has apeak in the vibration mode of the second order. The distribution of theparticipation factor Ki can parallelly be shifted toward a higher orderby shifting the number of natural vibrations of the winding toward itshigher order. The dotted curve in FIG. 14a shows the fact that a peakappears in the vibration mode of the third order, while the one-dotchain curve shows the fact that a peak appears in the vibration mode ofthe fourth order.

It will be understood that, according to the present invention, thevibration mode can be shifted toward a higher order by increasing thenumber of natural vibrations of the winding. It will be furtherunderstood that the same purpose can be attained by varying theelectromagnetic force distribution in a manner as described above so asto improve the vibration mode in order that the peak is shifted toward ahigher order. The latter state is shown in FIG. 14b. More precisely, thepeak of the participation factor Ki is shifted from its previousposition in the vibration mode of the second order to a position in thevibration mode of the fourth order so that consequently theparticipation factor corresponding to the vibration mode of the secondorder is reduced.

It will be understood from the above description that the displacementor force can be descreased as a whole by arranging in such a manner thatthe winding has a vibration mode of the fourth or higher order.

FIGS. 15 through 17 illustrate a few forms of a twowinding transformeremploying the above arrangement. In addition, although these drawingsonly schematically show a high-voltage winding, a low-voltage windingand an insulating material therefor by omitting detailed illustrationthereof, the construction of each portion in FIGS. 15 through 17 issimilar to that shown in FIG. 1. Referring to FIG. 15a, a low-voltagewinding 14 disposed closer to an iron core 1 is provided with asubstantially centrally disposed gap portion 14a having a large axialgap or a portion at which the number of turns is descreased to reducethe ampere-turn. A highvoltage winding 15 is disposed on the outside ofthe low-voltage winding 14. According to this arrangement, theelectromagnetic forces developed in the axial direction of the windingcan be distributed as shown in FIG. 15b so that the vibration mode ofthe fourth order has a node at a substantially central portion of thewinding. This arrangement is advantageous in that any appreciableexciting force is not imparted to the winding even when the windingresonates with the power source frequency or the frequency which isdouble the power source frequency in the vibration mode of lower orderand that, in the vibration mode of the fourth order, its number ofnatural vibrations is considerably high compared with the number ofexciting vibrations, resulting in a decrease in the displacement of thewinding due to vibration. Therefore, the electromagnetic forces thusdeveloped are distributed as shown by F F F and F and it is possible tominimize the resultant electromagnetic force.

Referring to FIG. 16a, a low-voltage winding 14 is substantiallytrisected to have portions 14b and which have a large axial gap, or thewinding 14 is provided with portions 14b and 140 at which the number ofturns is decreased to reduce the ampere-tum product. According to thisarrangement, the electromagnetic forces developed in the winding can bedistributed as shown in FIG. 16b so that the vibration mode of the sixthorder has two nodes adjacent to the trisecting points. Therefore, theelectromagnetic forces thus developed are distributed as shown by F,, FF F F and F and it is possible to minimize the resultant electromagneticforce.

Referring to FIG. 17a, a low-voltage winding 14 is provided adjacent toits quadrisecting points with portions 14d, 144? and 14f at which thenumber of turns is decreased to reduce the ampere-turn product.According to this arrangement, the electromagnetic forces developed invarious parts of the winding can be distributed in the vibration mode ofthe eighth order as shown by F through F in FIG. 17b, and it is possibleto minimize the resultant electromagnetic force. It will be understoodthat the electromagnetic force developed in the winding can be reducedby shifting the electromagnetic force distribution in the winding towarda vibration mode of higher order.

In the above embodiments, a tertiary winding may be arrangedconcentrically with respect to the leg portion of the iron core, thetertiary winding of these windings being such that the ampere-turnsthereof are reduced at respective local portions which are substantiallyequally spaced at predetermined distances from each other in the axialdirection of the tertiary winding.

FIGS. 18 through 21 show several preferred models for examining themechanical force developed in the axial direction of a transformerwinding.

FIG. 18 shows a model ofa two-winding transformer having concentricallyarranged windings and 21. A block of rigid ferromagnetic material suchas a laminated iron core 22 is disposed at the nodal point of vibrationof the inner winding 20 and outer winding 21, that is, at the point atwhich the magnetic flux distribution in the radial direction of thewindings is symmetri cal about the axis of the windings. The illustratedexample represents a 50 percent model employing solely the upper halvesof the windings. The model includes a transformer iron core 23,insulating rings 24 and 25 supporting the upper part of the innerwinding 20 and outer winding 21, respectively, with respect to thetransformer iron core 23, and an insulating base 26 supporting thelaminated iron core 22 on the lower part of the transformer iron core23.

The inner winding 20 is connected in series with the outer winding 21,and the direction of turns of the windings is so selected that theampere-turn of one winding is opposite to the ampere-turn of the otherwhen current is supplied to both these two windings. Therefore, when ashort-circuit current is supplied to one of these windings, the lines ofmain magnetic flux generated within the transformer iron core 23 canceleach other. Thus, the transformer iron core 23 may have a simplestructure which is sufficient to simulate only the leakage flux of eachwinding, and iron plates may be employed to construct a frame structurewhich suits the shape of the iron core.

FIG. 19 shows the distribution of the radial components of the leakageflux generated in the winding in the model shown in FIG. 18, that is,the distribution of the axial electromagnetic forces acting upon thewinding. By use of the above-described model, the state of generation ofelectromagnetic forces in a practical transformer winding caninexpensively be observed.

FIGS. 20 and 21 show models for examining the electromagnetic forcesgenerated in a sandwich arrangement of windings. The models comprise asimulation iron core 27, and a high-voltage winding 28 sandwichedbetween an upper low-voltage winding 29a and' a lower low-voltagewinding 29b. These windings are connected in series and are wound insuch a manner that their ampere-tums are opposite to each other as shownbyeBandSin the drawing. A block of rigid ferromagnetic material 30 isdisposed at the nodal point of vibration so that the model is equivalentto a practical transformer electro-magnetically as well as mechanically.Portions indicated by dotted lines in FIG. 20a show the images of thewindings.

Thus, according to the present invention, a model equivalent to apractical transformer can inexpensively be obtained and the examinationof the electromagnetic forces generated in the windings can be carriedout as if in the case of a practical transformer.

What is claimed is:

1. A short-circuit testing model for stationary induction apparatuseshaving a magnetic core comprising a leg portion and upper and lower yokeportions arranged above and below said leg portion, an inner windinghaving a plurality of turns disposed around the leg portion of themagnetic core, an outer winding having turns equal in number but reversein direction to said plurality of turns of said inner winding arrangedconcentrically around said inner winding and connected in seriestherewith, an insulating ring provided at one end of each of said innerand outer windings, and a ferromagnetic element provided at the otherend of each of said inner and outer windings.

2. A short-circuit testing model for stationary induction apparatusesaccording to claim 1, in which said insulating ring is separated foreach of the inner and outer windings, each of said separated insulatingrings being disposed to be opposite to one of the yoke portions of saidmagnetic core, and said ferromagnetic element is disposed to be oppositeto the other yoke portion opposite to said one yoke portion of saidmagnetic core through an insulating ring.

3. A short-circuit testing model for stationary induction apparatusesaccording to claim 2, in which'said ferromagnetic element is a unitaryblock which comprises laminated silicon steel plates and faces an endsurface of each of said inner and outer windings.

4. A short-circuit testing model for stationary induction apparatusesaccording to claim 1, in which the leg portion of said magnetic core isa solid ferromagnetic element having a frame construction.

1. A short-circuit testing model for stationary induction apparatuseshaving a magnetic core comprising a leg portion and upper and lower yokeportions arranged above and below said leg portion, an inner windinghaving a plurality of Turns disposed around the leg portion of themagnetic core, an outer winding having turns equal in number but reversein direction to said plurality of turns of said inner winding arrangedconcentrically around said inner winding and connected in seriestherewith, an insulating ring provided at one end of each of said innerand outer windings, and a ferromagnetic element provided at the otherend of each of said inner and outer windings.
 2. A short-circuit testingmodel for stationary induction apparatuses according to claim 1, inwhich said insulating ring is separated for each of the inner and outerwindings, each of said separated insulating rings being disposed to beopposite to one of the yoke portions of said magnetic core, and saidferromagnetic element is disposed to be opposite to the other yokeportion opposite to said one yoke portion of said magnetic core throughan insulating ring.
 3. A short-circuit testing model for stationaryinduction apparatuses according to claim 2, in which said ferromagneticelement is a unitary block which comprises laminated silicon steelplates and faces an end surface of each of said inner and outerwindings.
 4. A short-circuit testing model for stationary inductionapparatuses according to claim 1, in which the leg portion of saidmagnetic core is a solid ferromagnetic element having a frameconstruction.